We consider Toeplitz operators with unimodular symbols on the Hardy space H 2 on the unit circle. Examples of inner functions θ andareal function w such that ||w||∞ = π/2 and the Toeplitz operator with symbol θeiw is not left-invertible are given. We also study Toeplitz operators that are similar to isometries. Bibliography: 28 titles.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H. Berovii, Operator Theory and Arithmetic in H ∞, Amer. Math. Soc. Math. Surveys and Monographs, 26 (1988).
H. Berovii, “Notes on invariant subspaces,” Bull. Amer. Math. Soc., 23, 1–36 (1990).
A. Böttcher and S. M. Grudsky, “Toeplitz operators with discontinuous symbols: phenomena beyond piece-wise continuity,” Oper. Theory Adv. Appl., 90, 55–118 (1996).
A. Böttcher and B. Silbermann, Analysis of Toeplitz Operators, Springer (1990).
D. N. Clark, “On a similarity theory for rational Toeplitz operators,” J. Reine Angew. Math., 320, 6–31 (1980).
D. N. Clark, “On Toeplitz operators with loops,” J. Operator Theory, 4, 37–54 (1980).
D. N. Clark, “On Toeplitz operators with unimodular symbols,” Oper. Theory Adv. Appl., 24, 59–68 (1987).
D. N. Clark, “Perturbation and similarity of Toeplitz operators,” Oper. Theory Adv. Appl., 48, 235–243 (1990).
D. N. Clark and J. H. Morrel, “On Toeplitz operators and similarity,” Amer. J. Math., 100, 973–986 (1978).
P. L. Duren, Theory of HP Spaces, Academic Press (1970).
V. B. Dybin and S. M. Grudsky, Introduction to the Theory of Toeplitz Operators With Infnite Index, Oper. Theory Adv. Appl., 137 (2002).
M. F. Gamal’, “On Toeplitz operators similar to unilateral shift,” Zap. Nauhn. Sem. POMI, 345, 85–104 (2007).
M. F. Gamal’, “On Toeplitz operators similar to isometries,” J. Operator Theory, 59, 3–28(2008).
M. F. Gamal’, “On contrations that are quasiaffine transforms of unilateral shifts,” Acta Sci. Math. (Szeged), 74, 757–767 (2008).
R. Goor, “On Toeplitz operators which are contractions,” Proc. Amer. Math. Soc., 34, 191–192 (1972).
V. V. Peller, “When is a function of a Toeplitz operator close to a Toeplitz operator?” Oper. Theory Adv. Appl., 42, 59–85 (1989).
V. V. Peller, Hankel Operators and Their Applications, Springer Monographs in Math. (2003).
V. V. Peller and S. V. Khrushev, “Hankel operator, best approximation, and stationary Gaussian processes,” Usp. Mat. Nauk, 37, 53–124 (1982).
H. Radjavi and P. Rosenthal, Invariant Subspaces, Springer (1973).
M. Rosenblum and J. Rovnyak, Hardy Glasses and Operator Theory, Oxford Math. Monogr. (1985).
J. Rovnyak, “On the theory of unbounded Toeplitz operators,” Pacific J. Math., 31, 481–496 (1969).
D. Sarason, “Approximation of piecewise continuous functions by quotients of bounded analytic funtions,” Ganad. J. Math., 24, 642–657 (1972).
B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Spaces, North Holland, Amsterdam (1970).
K. Takahashi, “Injection of unilateral shifts into contractions,” Acta Sci. Math. (Szeged), 57, 263–276 (1993).
D. Wang, “Similarity theory of smooth Toeplitz operators,” J. Operator Theory, 12, 319–330 (1984).
D. V. Yakubovih, “Riemann surface models of a Toeplitz operator,” Oper. Theory Adv. Appl., 42, 305–415 (1989).
D. V. Yakubovih, “On the spectral theory of Toeplitz operators with a smooth symbol,” Algebra Analiz, 3, 208–226(1991).
D. V. Yakubovich, “Dual piecewise analytic bundle shift models of linear operators,” J. Funct. Anal., 136, 294–330(1996).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 376, 2010, pp. 5–24.
Rights and permissions
About this article
Cite this article
Gamal’, M.F. On Toeplitz operators with unimodular symbols: left invertibility and similarity to isometries. J Math Sci 172, 185–194 (2011). https://doi.org/10.1007/s10958-010-0191-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-010-0191-8